Proposition. Pullback of polynomial functions [fdip-002A]
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  • October 23, 2026

Let \(V\) be finite, \(J\subseteq V\), \(X=\mathbb R^V\) and \(Y=\mathbb R^J\). Let \(\pi :X\to Y\) restrict a configuration to \(J\). Write \(\mathcal P(X)\) and \(\mathcal P(Y)\) for the algebras of real polynomial functions generated by the coordinate evaluation functions, with pointwise operations and the identity involution. Then \[\begin {aligned} \pi ^*:\mathcal P(Y)&\longrightarrow \mathcal P(X),& P&\longmapsto P\circ \pi \end {aligned}\] is an injective unital real algebra map preserving these involutions. Every supplied real algebraic state on \(\mathcal P(X)\) therefore pulls back to one on \(\mathcal P(Y)\).

Proof. 

  • October 23, 2026

Each coordinate function \(y\mapsto y_j\), \(j\in J\), pulls back to \(x\mapsto x_j\). Composition preserves sums, products, real scalar multiples and constants by pointwise evaluation. Consequently a polynomial function pulls back to a polynomial function and \(\pi ^*\) is a unital real algebra map. Both involutions are the identity, so it preserves star. Define \(s:Y\to X\) by filling coordinates outside \(J\) with zero. Then \(\pi \circ s=\mathrm {id}_Y\). If \(P\circ \pi =Q\circ \pi \), composition with \(s\) gives \(P=Q\), proving injectivity. The state pullback proposition now applies to a supplied state. These arguments also cover \(J=\varnothing \) and \(V=\varnothing \); their real coordinate spaces are singletons, not empty sets.

Here polynomial functions are actual functions, rather than formal polynomial expressions. The proof uses coordinate composition and the zero-fill section, not a finiteness assertion about the real set \(X\). The algebra and positivity conventions are [stacks2017stacks, Section 22.2], Tag 09JF and [waldmann2004states, Definition 3.9, p. 11]. The finite-map construction gives the same contravariant direction, but its finite indicator expansion is not used here.

For the Gaussian expectation, measure and polynomial integrability first establishes that the Gaussian polynomial state on \(\mathcal P(X)\) exists. Once supplied, it restricts by the algebraic result above. Identifying its retained law requires the separate Gaussian marginal calculation. This proposition neither extends that analytic calculation to new degenerate cases nor asserts that all integrable functions form an algebra. Polynomial functions are also distinct from the algebra of field endomorphisms.